Stochastic Variance-Reduced Prox-Linear Algorithms for Nonconvex Composite Optimization - Zhang, Junyu; Xiao, Lin (Microsoft, 2020)
We consider minimization of composite functions of the form f(g(x))+h(x), where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an ε-stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity O(N+ N^[4/5]ε^[-1]) for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of O(ε^[-5/2]) and O(ε^[-3/2]) for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of O(N+N^[1/2]ε^[-1]) and O(ε^[-3/2]) are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations.